%!TEX root = thesis.tex
\chapter{Introduction}\label{chap:intro}

Loosely speaking, the essential dimension of an algebraic object is the minimal number of parameters needed to describe it.  It was introduced for finite groups by Buhler and Reichstein in \cite{BuhlerReichstein1997On-the-essentia}; for algebraic groups by Reichstein in \cite{Reichstein2000On-the-notion-o}; for algebraic stacks by Brosnan, Reichstein and Vistoli in \cite{BrosnanReichsteinVistoli2007Essential-dimen}; and for functors by Merkurjev in \cite{BerhuyFavi2003Essential-dimen}.

Essential dimension is of great interest in Galois cohomology, linear algebraic groups, and central simple algebras; has applications to Galois theory and the Noether problem; and is related to several open problems in algebra such as Albert's cyclicity conjecture, Hilbert's 13th problem, and Serre's Conjecture II.

The essential dimension of the symmetric group on $n$ letters, $\ed(S_n)$, has connections to the simplification of polynomials via Tschirnhaus transformations.  In Buhler and Reichstein's original paper \cite{BuhlerReichstein1997On-the-essentia}, their main interest was to determine how much a ``general polynomial of degree $n$'' can be simplified in this manner.  This may be viewed as a modern continuation of classical results of Hermite, Joubert, Klein and Hilbert.

The essential dimension of general finite groups is of interest in inverse Galois theory.  Here one wants to construct polynomials over a field $k$ with a given Galois group $G$.  Ideally, one wants polynomials that parametrize all fields extensions with that group: the so-called \emph{generic polynomials} (see \cite{KemperMattig2000Generic-polynom} and \cite{JensenLedetYui2002Generic-polynom}).  The essential dimension of $G$ is a lower bound for the \emph{generic dimension} of $G$: the minimal number of parameters possible for a generic polynomial.

This thesis will mainly consider finite groups of low essential dimension using birational geometry; we outline our approach in Section \ref{sec:EDviaBG}.  The major results of this thesis are a classification of all finite groups of essential dimension $2$ (see Section \ref{sec:ed2intro}), and a determination of the essential dimensions of the alternating group, $A_7$, and the symmetric group, $S_7$  (see Section \ref{sec:edS7intro}).

\section{Essential Dimension via Birational Geometry}\label{sec:EDviaBG}

Our study of essential dimension uses the concept of a \emph{versal $G$-variety} (defined in Chapter \ref{chap:prelim}).  These are simply models of the versal torsors seen in Galois cohomology.  The key fact we use is that if $G$ is a finite group of essential dimension $n$ then there exists a versal $G$-variety of dimension $n$.

This suggests an approach for classifying groups of essential dimension $n$: consider each faithful $G$-variety of dimension $n$ and determine whether it is versal.  Of course, enumerating all possible $G$-varieties of a given dimension is completely impractical.  We will see that it suffices to consider considerably smaller families.

For dimension $2$, we need only consider \emph{minimal rational $G$-surfaces}.  The minimal rational $G$-surfaces were classified by Manin \cite{Manin1967Rational-surfac} and Iskovskikh \cite{Iskovskih1979Minimal-models-} building on work by Enriques: they either possess conic bundle structures or they are del Pezzo surfaces.  The use of the Enriques-Manin-Iskovskikh classification for computing essential dimension was pioneered by Serre in his proof that $\ed_k(A_6)=3$ \cite[Proposition 3.6]{Serre2008Le-group-de-cre}.  Independently, Tokunaga \cite{Tokunaga2006Two-dimensional} has also investigated versal rational surfaces.

The dichotomy into conic bundle structures and del Pezzo surfaces is too coarse to easily identify exactly which groups occur.  Our current work was inspired by Dolgachev and Iskovskikh's \cite{DolgachevIskovskikh2006Finite-subgroup} finer classification of such groups.  Their goal was to classify conjugacy classes of finite subgroups of the Cremona group of rank $2$ (the group of birational automorphisms of a rational surface).  This problem has a long history.  The first classification was due to Kantor; an exposition of his results (with some corrections) can be found in Wiman \cite{Wiman1896Zur_Thorie_der_}.  Unfortunately, this early classification had several errors, and the conjugacy issue was not addressed.  More recent work on this problem include \cite{BayleBeauville2000Birational_invo}, \cite{deFernex2004On_planar_Cremo}, \cite{Zhang2001Automorphisms-o}, \cite{Beauville2007p-elementary-su} and \cite{Blanc2007Finite-abelian-}.

For higher dimensions, we are interested in \emph{unirational $G$-varieties}.  Unlike surfaces, we do not have powerful classification theorems.  However, Prokhorov recently used ideas from the minimal model program to classify finite simple groups which act faithfully on rationally connected threefolds \cite{Prokhorov2009Simple-finite-s}.  This work is crucial for our computation of the essential dimensions of $A_7$ and $S_7$.

\section{Finite Groups of Essential Dimension $2$}\label{sec:ed2intro}

Let $k$ be an algebraically closed field of characteristic $0$.  We use the notation $D_{2n}$ to denote the dihedral group of order $2n$.  Finite groups of essential dimension $1$ were classified by Buhler and Reichstein in their original paper; they are either cyclic or isomorphic to $D_{2n}$ where $n$ is odd.  There is a classification for infinite base fields by Ledet \cite{Ledet2007Finite-groups-o} (see also Remark \ref{rem:Ledet}), and for arbitrary base fields by Chu, Hu, Kang and Zhang \cite{ChuHuKangZhang2008Groups_with_ess}.

We review what is known about groups $G$ of essential dimension $2$.  If $G$ contains an abelian subgroup $A$ then $\rank(A) \le 2$.  The Sylow $p$-subgroups $G_p$ of $G$ can be described using the Karpenko-Merkurjev theorem \cite{KarpenkoMerkurjev2008Essential-dimen}: $G_p$ must be abelian for all $p$ odd, and groups $G_2$ must be of a very special form (see \cite[Theorems 1.2 and 1.3]{MeyerReichstein2008Some_consequenc}).  Any subgroup of $\GL_2(k)$ or $S_5$ has essential dimension $\le 2$.

Finite groups of essential dimension $2$ with non-trivial centres were classified (implicitly) by Kraft, L\"otscher and Schwarz (see \cite{KraftSchwarz2007Compression-of-} and \cite{KraftLotscherSchwarz2009Compression-of-}).  They show that a finite group with a non-trivial centre has essential dimension $\le 2$ if and only if it can be embedded in $\GL_2(k)$.  Their main interest was in \emph{covariant dimension}, a ``regular'' analog of essential dimension.  See also \cite{Reichstein2004Compressions-of} and \cite{Lotscher2008Application-of-}.

Recall that the automorphism group of the algebraic group $(k^\times)^n$ is isomorphic to $\GL_n(\bbZ)$.  Our main theorem is as follows:

\begin{thm}\label{thm:ed2classification}
Let $T = (k^\times)^2$ be a $2$-dimensional torus.  If $G$ is a finite group of essential dimension $2$ then $G$ is isomorphic to a subgroup of one of the following groups:
\begin{enumerate}
\item $\GL_2(k)$, the general linear group of degree $2$,
\label{thm:ed2:GL2}
\item $T \rtimes \calG_1$ with $|G \cap T|$ coprime to $2$ and $3$\\
$\calG_1 = \left\langle \mat{1&-1\\1&0}, \mat{0&1\\1&0} \right\rangle \simeq D_{12}$,
\label{thm:ed2:G1}
\item $T \rtimes \calG_2$ with $|G \cap T|$ coprime to $2$\\
$\calG_2 = \left\langle \mat{-1&0\\0&1}, \mat{0&1\\1&0} \right\rangle \simeq D_{8}$,
\label{thm:ed2:G2}
\item $T \rtimes \calG_3$ with $|G \cap T|$ coprime to $3$\\
$\calG_3 = \left\langle \mat{0&-1\\1&-1}, \mat{0&-1\\-1&0} \right\rangle \simeq S_3$,
\label{thm:ed2:G3}
\item $T \rtimes \calG_4$ with $|G \cap T|$ coprime to $3$\\
$\calG_4 = \left\langle \mat{0&-1\\1&-1}, \mat{0&1\\1&0} \right\rangle \simeq S_3$,
\label{thm:ed2:G4}
\item $\PSL_2(\bbF_7)$, the simple group of order $168$,
\label{thm:ed2:PSL27}
\item $S_5$, the symmetric group on $5$ letters.
\label{thm:ed2:S5}
\end{enumerate}
Furthermore, any finite subgroup of these groups has essential dimension $\le 2$.
\end{thm}

The proof of Theorem \ref{thm:ed2classification} breaks into two mostly independent pieces.  We show that it suffices to consider only four surfaces:

\begin{thm}\label{thm:ed2to4surfaces}
If $G$ is a finite group of essential dimension $2$ then $G$ has a versal action on one of the following: the projective plane $\bbP^2$, the product of projective lines $\bbP^1 \times \bbP^1$, or a del Pezzo surface of degree $5$ or $6$.
\end{thm}

Then, we show that the groups with versal actions on these four surfaces are those listed in Theorem \ref{thm:4surfacesClassification} above.

We also mention some intermediate results that we feel are of independent interest.  Many interesting versal varieties are toric.  In order to classify versal actions on toric varieties, we develop techniques that apply to smooth complete toric varieties of arbitrary dimension.  Our major tools are the theory of \emph{Cox rings} \cite{Cox1995The-homogeneous} and \emph{universal torsors} \cite{ColliotTheleneSansuc1987La_descente_sur}.  The main result on toric varieties is as follows (Theorem \ref{thm:VersalCoxSplit}): a faithful $G$-action on a complete non-singular toric variety is versal if and only if it lifts to an action on the variety of the associated Cox ring.

This result has some important corollaries.  First, if a complete non-singular toric variety has a $G$-fixed point then it is versal (Corollary \ref{cor:fixedPointImpliesVersal}).  Second, a complete non-singular toric variety is $G$-versal if and only if it is $G_p$-versal for all of its $p$-subgroups (Corollary \ref{cor:versalByPGroups}).  This second corollary is instrumental in our proof of Theorem \ref{thm:4surfacesClassification}.

\section{Essential Dimension of $A_7$ and $S_7$}\label{sec:edS7intro}

The essential dimensions of the alternating groups, $A_n$, and the symmetric groups, $S_n$, are of special interest because they relate to classical questions of simplifying degree $n$ polynomials via Tschirnhaus transformations.  In particular, the degree $7$ case features prominently in algebraic variants of Hilbert's 13th problem.  In this language, several results for small $n$ were established by Hermite, Joubert and Klein in the 1800s. For more information, see Chapter \ref{chap:conclusion}.  The values of $\ed_k(S_n)$ and $\ed_k(A_n)$ are known for all $n \le 6$; see \cite{BuhlerReichstein1997On-the-essentia} and \cite[Proposition 3.6]{Serre2008Le-group-de-cre}.

The second major result of this thesis is the following:

\begin{thm}\label{thm:edS7}
Let $k$ be a field of characteristic $0$ (not necessarily algebraically closed).
Then $\ed_k(A_7)=\ed_k(S_7)=4$.
\end{thm}

The proof relies on recent work of Prokhorov \cite{Prokhorov2009Simple-finite-s} on the classification of rationally connected threefolds with faithful actions of non-abelian simple groups.

\section{Overview}

The rest of the thesis is structured as follows.

In Chapter \ref{chap:prelim}, we recall basic facts about versal varieties, essential dimension and the Enriques-Manin-Iskovskikh classification.  In Chapter \ref{chap:toricVersal}, we develop tools for determining when a toric $G$-variety is versal.  

Chapters \ref{chap:4surfaces} through \ref{chap:DelPezzo} are devoted to finite groups of essential dimension $2$. In Chapter \ref{chap:4surfaces}, we determine precisely which groups act versally on the four surfaces of Theorem \ref{thm:ed2to4surfaces}.  In Chapter \ref{chap:conics}, we show that all groups acting versally on conic bundle structures already act versally on the four surfaces.  In Chapter \ref{chap:DelPezzo}, we show the same for the del Pezzo surfaces. This proves Theorem \ref{thm:ed2to4surfaces} and, thus, Theorem \ref{thm:ed2classification}.

Chapter \ref{chap:edS7} is a proof of Theorem \ref{thm:edS7}.

Finally, Chapter \ref{chap:conclusion} discusses applications of these results, and directions for future research.
